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Non-classical, bounded Fechnerian integration for loudness: contrary to Luce and Edwards, initial loudness-difference-size stipulations are only recouped for linear loudness growth

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A major question in sensory science is how a sensation of magnitude F (such as loudness) depends upon a sensory stimulus of physical intensity I (such as a sound-pressure-wave of a particular root-mean-square sound-pressure-level). An empirical just-noticeable sensation difference (ΔF)j at Fj specifies a just-noticeable intensity difference (ΔI)j at Ij. Intensity differences accumulate from a stimulus-detection threshold Ith up to a desired intensity I. Likewise, the corresponding sensation differences are classically presumed to accumulate, accumulating up to F(I) from F(Ith), a non-zero sensation (as suggested by hearing studies) at Ιth. Consequently, sensation growth F(I) can be obtained through Fechnerian integration. Therein, empirically-based relations for the Weber Fraction, ΔI/I, are individually combined with either Fechner’s Law ΔF = Β or Ekman’s Law (ΔF/F) = ɡ; the number of cumulated steps in I is equated to the number of cumulated steps in F, and an infinite series ensues, whose higher-order terms are ignored. Likewise classically ignored are the integration bounds Ith and F(Ith). Here, we deny orthodoxy by including those bounds, allowing hypothetical sensation-growth equations for which the differential-relations ΔF(I) = F(I+ ΔI) - F(I) or (ΔF(I)/F(I)) = (F(II) - F(I))/F(I) do indeed return either Β or ɡ, for linear growth of sensation F with intensity I. Also, 24 sensation-growth equations F(I), which had already been derived by the author likewise using bounded Fechnerian integration (12 equations for the Weber Fraction (ΔI/I), each combined with either Fechner’s Law or with Ekman’s Law), are scrutinized for whether their differential-relations return either Β or ɡ respectively, particularly in the limits (ΔI/I) << 1 and the even-more-extreme limit (ΔI/I) → 0, both of which seem unexplored in the literature. Finally, some relevant claims made by Luce and Edwards (1958) are examined under bounded Fechnerian integration: namely, that three popular forms of the Weber Fraction, when combined with Fechner’s Law, produce sensation-growth equations that subsequently return the selfsame Fechner’s Law. Luce and Edwards (1958) prove to be wrong.

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